Polyhedra and the surface of a prism
Answers are for parents and teachers.
1
A convex polyhedron has 6 square faces and 8 equilateral-triangle faces; every edge belongs to two faces. Find the numbers of edges and vertices.
E = (6 · 4 + 8 · 3)/2 = 24; V = 2 − F + E = 2 − 14 + 24 = 12.
2
A convex polyhedron whose faces are all triangles has 12 faces. Find the numbers of edges and vertices. (Every edge belongs to two faces.)
3F = 2E ⇒ E = 18; V = 2 − 12 + 18 = 8.
3
A regular quadrangular prism has base side 3 and diagonal 11. Find its height and total surface. (The base diagonal is 3√2.)
d² = 18 + h² = 121 ⇒ h² = 103, h = √103 ≈ 10.15; S = 2 · 9 + 4 · 3 · h = 18 + 12√103 ≈ 139.8.
4
The three faces at one vertex of a cuboid have areas 10, 15 and 6. Find its dimensions.
ab = 10, bc = 15, ac = 6 ⇒ (abc)² = 900, abc = 30 ⇒ c = 3, a = 2, b = 5. The dimensions are 2, 5, 3.
5
An oblique triangular prism has lateral surface 180 cm² and lateral edge 12 cm. Its perpendicular section is an equilateral triangle. How many centimetres is its side?
5